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Complex transformation method and resonances in one-body quantum systems. (English) Zbl 0568.47008

The author treats the resonance problem in one-body systems via a new spectral deformation method. A third complex transformation method is given which may be compared to the local distortion technique of Babbitt and Balslev and the exterior complex scaling method of Simon. As with these latter methods the idea is to extend the domain of application of the complex scaling method of Combes and Balslev in the study of resonances. His method generalizes the complex scaling method in a way which is close in spirit to the local distortion technique. It is applicable to the multicenter problems in which each potential can be represented more or less as a sum of exponentially decaying and dilation analytic spherically symmetric parts. In an addendum to his paper the author notes that condition 3.8 has be modified.
Reviewer: M.Thompson

MSC:

47A40 Scattering theory of linear operators
81U10 \(n\)-body potential quantum scattering theory
35P25 Scattering theory for PDEs
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References:

[1] J. Aguilar and J.M. Combes , On a class of analytic perturbations of one body Schrödinger operators , Commun. Math. Phys. , t. 22 , 1971 , p. 269 - 279 . Article | MR 345551 | Zbl 0219.47011 · Zbl 0219.47011
[2] D. Babbitt and E. Balslev , A characterization of dilation-analytic potentials and vectors , J. Funct. Anal. , t. 18 , 1975 , p. 1 - 14 . MR 384008 | Zbl 0304.47009 · Zbl 0304.47009
[3] D. Babbitt and E. Balslev , Local distortion technique and unitarity of the S-matrix for the 2-body problem , J. Math. Anal. Appl. , t. 54 , 1976 , p. 316 - 347 . MR 413860
[4] E. Balslev , Local spectral deformation techniques for Schrödinger operators , Mittag-Leffler preprint , 1982 . MR 756770
[5] E. Balslev and J.M. Combes , Spectral properties of Schrödinger Hamiltonians with dilation analytic potentials , Commun. Math. Phys. , t. 22 , 1971 , p. 280 - 294 . Article | MR 345552 | Zbl 0219.47005 · Zbl 0219.47005
[6] E.A. Coddington and N. Levinson , Theory of Ordinary Differential Equations , McGraw-Hill , 1955 . MR 69338 | Zbl 0064.33002 · Zbl 0064.33002
[7] J.M. Combes , An algebraic approach to quantum scattering , 1969 , unpublished manuscript. · Zbl 0181.27604
[8] W. Hunziker , The Schrödinger eigenvalue problem for N-particle systems , in The Schr&dinger Equation , W. Thrrring and P. Urban, eds., Springer-Verlag , 1977 . Zbl 0365.47025 · Zbl 0365.47025
[9] A. Jensen , Local distortion technique, resonances and poles of the S-matrix , J. Math. Anal. Appl. , t. 59 , 1977 , p. 505 - 513 . MR 441153 | Zbl 0361.47018 · Zbl 0361.47018
[10] A. Jensen , Resonances in an abstract analytic scattering theory , Ann. Inst. H. Poincaré , t. 33 , 1980 , p. 209 - 223 . Numdam | MR 605196 | Zbl 0462.47010 · Zbl 0462.47010
[11] E. Mourre , Absence of singular spectrum for certain self-adjoint operators , Commun. Math. Phys. , t. 78 , 1981 , p. 391 - 408 . Article | MR 603501 | Zbl 0489.47010 · Zbl 0489.47010
[12] C.W. Mccurdy , Jr. and T.N. Rescigno , Extension of the method of complex basis functions to molecular resonances , Phys. Rev. Lett. , t. 41 , 1978 , p. 1364 - 1368 .
[13] J. Nuttall , Analytic continuation of the off-energy-shell scattering amplitude , J. Math. Phys. , t. 8 , 1967 , p. 873 - 877 .
[14] M. Reed and B. Simon , Methods of Modern Mathematical Physics, I, II, IV , Academic Press . MR 751959 · Zbl 0401.47001
[15] Sanibel WORKSHOP on Complex Scaling , 1978 , in Int. J. Quant. Chemistry , t. 14 , 1978 .
[16] I.M. Sigal , Scattering Theory for Many-Body Quantum Mechanical Systems-Rigorous Results , Springer , Lecture Notes in Mathematics N1011 , 1983 . MR 715786 | Zbl 0522.47006 · Zbl 0522.47006
[17] I.M. Sigal , A generalized Weyl theorem and the Lp-spectra of Schrödinger operators , J. Oper. Theory (to appear). MR 768306 | Zbl 0582.47003 · Zbl 0582.47003
[18] B. Simon , Quadratic form technique and the Balslev-Combes theorem , Commun. Math. Phys. , t. 27 , 1972 , p. 1 - 9 . Article | MR 321456 | Zbl 0237.35025 · Zbl 0237.35025
[19] B. Simon , Resonances and complex scaling: A rigorous overview , J. Quant. Chem. , t. 14 , 1978 , p. 529 - 542 .
[20] B. Simon , The definition of molecular resonance curves by the method of exterior complex scaling , Phys. Lett. , t. 71 A, 1979 , p. 211 - 214 .
[21] L. Thomas , On the spectral properties of some one-particle Schrödinger Hamiltonians , Helv. Phys. Acta , t. 45 , 1972 , p. 1057 - 1065 . MR 376023
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